Question

A geometric sequence can be used to describe the population of rabbits on a

farm. The first spring, the farmer purchased 8 rabbits. Five years later, there are
648 rabbits at the farm. Assuming that none of the rabbits leave the farm, how
many rabbits were on the farm in year 3?

Answer 1

The population of rabbits on the farm in year 3 was 72, calculated using the formula for the nth term in a geometric sequence after finding the common ratio to be 3.

Step-by-step explanation:

To determine how many rabbits were on the farm in year 3, we need to recognize that the population's growth can be described by a geometric sequence. The first term in this sequence is 8 rabbits, and, after five years, the population increased to 648 rabbits. The general form of a geometric sequence is given by:

an = a1 × rn-1

Here, an is the population in the nth year, a1 is the initial population, r is the common ratio, and n is the number of terms. If we take the fifth year as the fifth term, we can calculate the common ratio as follows:

648 = 8 × r5-1

r4 = 81

r = 3

With the common ratio found, we can calculate the population in year 3, which corresponds to the third term (a3) of the sequence:

a3 = 8 × 33-1

a3 = 8 × 9

a3 = 72

Therefore, there were 72 rabbits on the farm in year 3.

Answer 2

step 1

Find the average growth per year of the populations of rabbits farm A

Over the first 2 years

for t=0

numbers of rabbits=5 (is not exact)

for t=2

numbers of rabbits=40

average=[40-5]/2---------> 17.5

step 2

Find the average growth per year of the populations of rabbits farm B

Over the first 2 years

for t=0

numbers of rabbits=5 (is not exact)

for t=2

numbers of rabbits=30

average=[30-5]/2---------> 12.5

step 3

Compare the average growth per year of the populations of rabbits on both farms

farm A=17.5

farm B=12.5

the average rate of population growth of rabbits in farm A is greater than average rate of population growth of rabbits in farm B by about 5 rabbits per year.

therefore

the answer is the option C)

the average rate of population growth of rabbits in farm A is greater than average rate of population growth of rabbits in farm B by about 6 rabbits per year.

Step-by-step explanation: